Bad news unfortunately! While I was struggeling with my MacBook, I checked my DiskUtility again and the massage was (something like): "There has been a fatal hard disk problem. Try to back-up what you can and replace the HD".

I spent hours and hours to back-up everything (I could only do it in small peaces) and then I brought my notebook to the Apple-store. They were surprised that there was a fatal HD-problem and in the meantime de DiskUtility-program could still tell me that there was a problem. And that I could still back-up in a way.

Now, it has been sent away to be repaired. I have to wait for about a week (or 2) and then I hope everything is fixed...

Thanks for your help or at least constatating that it was a HD problem. I don't hope that you see me back here soon with the same kind of problem .